Questions tagged [cardinals]

This tag is for questions about cardinals and related topics such as cardinal arithmetics, regular cardinals and cofinality. Do not confuse with [large-cardinals] which is a technical concept about strong axioms of infinity.

Filter by
Sorted by
Tagged with
1
vote
0answers
22 views

Is the set of irrational numbers smaller than the set of real numbers? [duplicate]

I am not a mathematician, but I study it for fun. I know that Cantor showed that the infinite set of the real numbers is larger than the rationals since the real numbers are uncountable and the ...
1
vote
0answers
39 views

a name for a class of cardinals

I'm not a native English speaker; what S in SCar below might stand for ?
1
vote
0answers
68 views

Proving that $\left|A \times A\right|$ is equal to $\left|A\right|$ for every infinite set

How do you prove that $\left|A \times A\right|$ is equal to $\left|A\right|$ for every infinite set? I'm trying to prove this basic fact of cardinal arithmetic, but I'm getting stuck on the ...
1
vote
1answer
48 views

Is there an intuitive way of justifying why the square of an infinite cardinal is itself?

By no means I am an expert in this subject, but I do have some knowledge of ZFC. While there are many proofs which are difficult to recollect, I feel like I have enough knowledge that if I am given a ...
1
vote
1answer
44 views

How to prove $a^x\times a^y=a^{x+y}$ for cardinals? [duplicate]

How can I prove this: $$a^x\times a^y=a^{x+y} $$ when $card(A)=a$ , $card(X)=x$ and $card(Y)=y$.
4
votes
0answers
90 views

Proof with transfinite induction

I'm trying to prove the following statement: Suppose that for every $r\in\mathbb{R}$ we are given a finite set $A_r\subseteq\mathbb{R}$ and that for any finite set $D\subseteq\mathbb{R} $, there ...
0
votes
0answers
30 views

Existence of a “direct union of subsets” operation (matching direct sum of vector subspaces)?

Dimension of vector space and cardinal number of sets have very similar properties : Let $U$ be a vector space and $V, W \subseteq U$ two vector subspace, so $$ \dim(V + W) = \dim(V) + \dim(W) - \dim(...
2
votes
2answers
95 views

Proof that $ \mathbb{R} $ is uncountable

Im sure this question has been asked here a lot, but I'd like to hear if the way I understood Cantor's diagonal proof is correct. We know that $ \left(0,1\right)\sim\mathbb{R} $. So its enough to ...
1
vote
1answer
97 views
+50

Find the cardinality of $\big\{(x,y,z)\mid x^2+y^2+z^2= 2^{2018}, xyz\in\mathbb{Z} \big\}$.

What is the cardinality of set $\big\{(x,y,z)\mid x^2+y^2+z^2= 2^{2018}, xyz\in\mathbb{Z} \big\}$? Since I have very limited knowledge of number theory , I tried using logarithms and then ...
-2
votes
0answers
51 views

Show that a set is countable. [closed]

Let $A=\{ \frac{\sqrt[m]{m}}{n^n} : m,n \in \mathbb{N} \}$. I can't find a bijection between A and $\mathbb{N}$. Any tip with the function? Thanks.
2
votes
0answers
28 views

Not a correct subscript?

In his book Proper and Improper forcing on page $42$ in the proof of Fact 7.3 on line $4$ Shelah says "where ${\cal S_{\leq \aleph_1}}(A)$" but he never uses this expression. He only uses it ...
1
vote
2answers
56 views

I solved but don't know if it is correct, can you help me? Showing $P(X\cup Y)\approx P(X)\times P(Y)$

Question: $X, Y$ are infinite sets that are not empty, and $X\cap Y=\emptyset$. Show $P(X\cup Y)\approx P(X)\times P(Y)$ Hi! I tried to solve the question I wrote above, but I don't know if it is ...
0
votes
1answer
40 views

Ratio of two infinite cardinal numbers

Suppose $G$ is the group of all functions between $[0,1]\to\mathbb{Z}$. Let $H$ be the subgroup defined as $H=\{f\in G: f(0)=0\}$. Then, what can be said about the cardinality of $H$ and its index in $...
0
votes
2answers
47 views

How to choose infinite number of different values from infinite set of infinite sets.

Let $ \aleph_{\alpha} $ be a cardinal and assume that $ \left\{ A_{\beta}:\beta<\aleph_{\alpha}\right\} $ is a set of sets, such that $ |A_{\beta}|=\aleph_{\alpha} $ for any $ \beta<\aleph_{\...
0
votes
3answers
24 views

Prove that any countable cartesian product of countable sets is countable [duplicate]

I want to prove that infinite (yet countable) cartesian product of countable sets is countable. Here's what I tried: Step 1: I proved that for 2 countable sets $ A_1,A_2 $ , the product $ A_{1}\times ...
1
vote
1answer
27 views

Hartogs set of a well ordered set

so i understood that the hartogs set of a well ordered set $A$ is defined as $H(A)$ the minimal ordinal such that $H(A)\nleq A$ (there is no injection from $H(A)$ to $A$) and i also uderstood the ...
0
votes
1answer
34 views

Prove that negation of the continuum hypotheses implies existence of subset of R such that…

Prove that the negation of the continuum hypothesis implies that there exist $A⊂R$ such that $ℵ_0<|A|<|R|$. The negation of the hypotheses implies existence of a set B such that $ℵ_0<|B|<|...
1
vote
2answers
50 views

Prove that $ \kappa\times\lambda=\lambda $

let $ \kappa<\lambda $ and assume $ \aleph_{0}\leq\lambda $ prove that: $ \kappa\times\lambda=\lambda $ So, my attempt, based on the fact that i already proved for infinite cardinals $ \lambda $ ...
1
vote
1answer
16 views

How to convert degrees of a circle into positive and negative X and Y values

I am hoping this is a very simple equation that can solve this (I am not a mathematician) but considering I have values in degrees (0 - 360), I am trying to convert that back into plus and minus X and ...
1
vote
1answer
32 views

Can i prove the |P(Z+)}=|(0,1)| list like that?

First of all,in diagonalization prove,we can always generate the a real number that not in the list by adding one to the value of the diagonal. In power set of positive integer,if we use the real ...
1
vote
1answer
67 views

Axiom of Choice in $\kappa \cdot \kappa = \kappa$

There's this proof that $\kappa \cdot \kappa = \kappa$ for $\kappa\in\text{Card}$ infinite by induction and well-ordering $\kappa \times \kappa$ with: $(\alpha,\beta)<^*(\alpha',\beta')\iff\max(\...
0
votes
1answer
42 views

Is the power of set of elements that are less than $x$ little?

Does every set $X$ can be well ordered such that for all element $x$ the power of set of elements that are less than $x$ is less than the power of $X$? I saw this idea in proof of some problem. Can ...
1
vote
1answer
38 views

Prove that any 2 bases of a vector space has the same cardinality

I know this question has been asked before, but I tried to prove it myself and I cant finish my prove because im not sure how to write the contradiction in a foraml and correct way. Let V be a vector ...
0
votes
1answer
28 views

Prove that f(m,n)=(m+n−2)(m+n−1)/2+m from $\Bbb Z^+\times \Bbb Z^+$→ $\Bbb Z^+$ is one-to-one.(2)

Since the last post no one gave the solution, so i reopen one and use other approach searched in this forumn. https://math.stackexchange.com/a/91323/620871 Show that the polynomial function $$f(m,n)=(...
0
votes
1answer
38 views

What is aleph-one to the power of aleph-null

What is $\aleph_1^{\aleph_0}$? Can anyone shed some light. I'm not sure if this is provable or even has a value. Thanks
-2
votes
0answers
14 views

The set of polynomials with $n$ variables and coefficients in a countable set is countable

I'd like to know whether the set of polynomials $S[X_1, \dotsc, X_n]$ with coefficients in a countable set $S$ is countable. I think it must be, but I don't know how to prove it. Thanks.
1
vote
0answers
71 views

Prove that $f(m,n)=\frac{(m+n−2)(m+n−1)}{2}+m$ from $\mathbb{N}^2\to\mathbb{N}$ is one-to-one.

Show that the polynomial function $$f(m,n)=(m+n−2)(m+n−1)/2+m $$ is one-to-one and onto. Both domain is $\Bbb Z^+\times \Bbb Z^+$, codomain are $\Bbb Z^+$. I want to prove $f(a,b)=f(c,d) \...
0
votes
0answers
35 views

How to prove that the cardinality of Collection of Borel subsets is equal to cardinality of real number?

One way is obvious. Since every point set of R is a Borel set, so the Collection of Borel sets have at least the cardinality of R. I struggled to find an injection from The class of Borel subset to R? ...
0
votes
0answers
28 views

Proving that a successor cardinal $k^+$ is regular, given that $k$ is an infinite cardinal and using a different definition of cofinality

I have been trying to prove that if $k$ is an infinite cardinal number, then its successor $k^+$ is regular, that is $\operatorname{cf}(k^+)=k^+$. In my class we have defined cofinality in the ...
-1
votes
1answer
25 views

Uncountability of $\{x \in [0,1]: f(x)\geq \epsilon \}$ where $f:[0,1]\rightarrow (0,\infty)$. [closed]

Let $f:[0,1]\rightarrow (0,\infty)$. Prove there exists some $\epsilon>0$ such that $\{x \in [0,1]:f(x)\geq \epsilon\}$ is uncountable.
0
votes
1answer
24 views

$ (\prod_{i\in I}\kappa_i)^\mu = \prod_{i\in I}\kappa_i^\mu $, where $ \kappa_i ,\mu $ are infinite cardinals, $I$ an infinite set.

If $|B|=\mu$ and $|A_i|=\kappa_i \;\forall i\! \in\! I$, than $(\prod_{i\in I}\kappa_i)^\mu =|\text{Fun}(B,\prod_{i \in I}A_i)|$. Also, $ \prod_{i\in I}\kappa_i^\mu = |\prod_{i \in I}{|\text{Fun}(B,...
0
votes
2answers
22 views

Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$.

Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$. The subgroup $H= \left \{ f \in G :f(0)=0 \right \}$ Then $1)$ $H$ is countable $2)$ $H$ is uncountable $3)$ $H$ ...
1
vote
1answer
35 views

Explanation of Cardinal Arithmetic Used in Proving that bases of Vector spaces have the same cardinality.

Let $V$ be a vector space with bases $B_1$, $B_2$. For all $b\in B_1$ there exists $U_b\subset B_2$ such that $U_b$ is finite and $b\in span(U_b)$. Hence, $V=span(B_1)=span(\cup_{b\in B_1}U_b)$. Since ...
2
votes
1answer
37 views

Given $\kappa = \sup_{\alpha< \lambda} \kappa_{\alpha}$n can we assume the $\{\kappa_\alpha: \alpha < \lambda\}$ is strictly increasing?

Suppose $$\kappa= \sup_{\alpha < \lambda} \kappa_\alpha$$ where $\kappa$ is an infinite cardinal and $\kappa_\alpha$ are cardinals, $\lambda$ is a non-zero limit ordinal, $\lambda < \kappa$ and $...
1
vote
1answer
40 views

Proving that ${\aleph_1}^{\aleph_0}\leq |[\omega_1]^{\omega}|$

Today I was working with some exercises about topology but in some part I need to prove the next inequality: $${\aleph_1}^{\aleph_0}\leq |[\omega_1]^{\omega}|$$Here $[\omega_1]^{\omega}:=\left\{A\...
1
vote
1answer
27 views

$(\lambda,D)$-model homogenity

Here on the page $41$ in the definition $6.1(3)$ I do not follow where $D$ from $6.1(3)$ appears in the definition of $(\lambda,D)$-model homogenity in $6.1(2)$. It appears in the first $2$ paragraphs ...
0
votes
1answer
21 views

Unknown index in a model theoretic considerations

What is $k$ in $${}^kM$$ here on the page $11$ in the definition $2.10$?
3
votes
1answer
136 views

Forcing $2^{\omega} = \omega_{\omega_1}$ together with $2^{\omega_1} = \omega_{\omega_2}$

I'm studying from Kunen's Set Theory, and I came across this exercise (G6 ch. 7): Suppose $M$ satisfies $\text{GCH}$. Let $\kappa_1 < \dots < \kappa_n$ be regular cardinals of $M$ and let $\...
0
votes
1answer
23 views

Let S, T and P be three nonempty set. Prove that (a)S~S (b)If S~T, then T~S

(a) S~S means it is reflexive (b) If S~T, then T~S means it is symmetry Using the definition of equivalent sets, set S is equivalent to T if and only if there exists a function f:S->T which is one-to-...
-1
votes
1answer
19 views

Suppose that $S\sim T$, $P\sim Q$, $S⋂P=\varnothing $ and $T⋂Q=\varnothing$. Prove that $(S∪P) \sim (T∪Q)$. [closed]

By definition of equivalent sets, a set $S$ is equivalent to set $T$ if and only if the function $f:S\to T$ is one-to-one and onto. A set $P$ is equivalent to set $Q$ if and only if the function $f:P\...
2
votes
2answers
48 views

Uncoutable set minus a countable subset is uncountable without axiom of choice or continuum hypothesis [duplicate]

Is there a way of proving that if $|X|=2^{\aleph_0}$ and $Y\subseteq X$ such that $|Y|=\aleph_0$ then $|X-Y|=2^{\aleph_0}$ without using axiom of choice or continuum hypothesis. Most of the proofs I'...
4
votes
0answers
40 views

Which sets have nontrivial elementary extensions (with respect to all possible relations) of the same cardinality?

Let $\kappa$ be an infinite cardinal, and take a set $X$ of cardinality $\kappa$. Consider $X$ as a first-order structure with respect to the language that consists of all possible finitary relations ...
1
vote
1answer
36 views

Let $\alpha$ be a limit ordinal. $f: \alpha \to \beta $ be strictly increase cofinal map. then $cf(\alpha) = cf(\beta)$.

Let $\alpha$ be a limit ordinal. $f: \alpha \to \beta $ be strictly increase cofinal map.then $cf(\alpha) = cf(\beta)$. refer to proof in the Kenneth Kunen's book. It's easy to show that $cf(\alpha) ...
0
votes
3answers
61 views

Bijection between real and natural numbers.

I know, I know, this question has been asked several times, but I feel mine is a little bit different. Imagine a correspondence between $[0,1]$ and natural numbers in the following sense: $$ 0.12 \...
0
votes
1answer
34 views

Cardinal of quotient set

Consider a finite set $X$ and let's denote by $F$ the set of all maps defined on $X$ with values in $\mathbb{N}$. How to prove that $F$ is countable. Let $\pi_1, \pi_2 \in F$. We define the binary ...
1
vote
0answers
41 views

Find the Cardinality of points on a circle

$n$ points are distributed on the perimeter of a circle. We connect each $2$ points with a straight line, and we get the set $S$ of the chords. (1) Find $|S|$ (2) Assuming that there are no inner ...
6
votes
2answers
132 views

How should I self-study set theory/cardinality?

So, I am an absolute beginner in mathematics; only being knowledgeable in some basic ideas of the subject. My interest in math started only recently, while reading about set theory and cardinality (...
-1
votes
1answer
37 views

Prove |A⋃B| = |A| + |B - A| [closed]

I'm not really sure how to start this. I considered proving that there is a bijection between A⋃B and A⋃(B-A) but I'm struggling finding a 1-1 function f: A⋃B -> A⋃(B-A) so that A⋃B <= A⋃(B-A) or ...
0
votes
1answer
32 views

Exponentiation of cardinal numbers [duplicate]

How to prove that $a^{x}\leq b^{y}$ for any cardinal number a,b, $x,y$ with a$\leq$b , $x\leq y$? First, I want to show that $p^r\leq\ q^r$ and $r^p\leq\ r^q$ for any cardinal number $p,q,r$ with $p\...
20
votes
2answers
977 views

Why does the number of possible probability distributions have the cardinality of the continuum?

Wikipedia's article on parametric statistical models (https://en.wikipedia.org/wiki/Parametric_model) mentions that you could parameterize all probability distributions with a one-dimensional real ...

1
2 3 4 5
56